198 XIII INTEGRATION The last assertion follows from the definition of equivalent measures. To prove the first assertion, we may, by virtue of (13.15.7), assume that the measures un are bounded. By multiplying each jun by a suitable positive real number, we may also assume that jun(l) :g 2~M. If vn = sup(jKl9//2, ... , /O> then by virtue of (13.15.3.2) we have vn(f) g £/4(/) g 1 for every /e JfR(X) such that 0 g/^ 1, and every n. k-l Hence (13.4.4) the increasing sequence (vn) converges vaguely to its least upper bound v in MR(X), and we have v(/) :g 1 for every /e Jf R(X) such that 0 ^/^ 1. Hence v*(l) g 1 and so v is bounded. We may therefore write l*n— 9n' v> where #„ is locally v-integrable and ^0; hence vn — i sup gk\ • v \l£**n / (13.15.3.3) and consequently v=(sup#w\-v (13.15.3.5). It is now clear \ n / (1 3.15.5) that the relation v(N) = 0 implies that nn(N) = 0 for all n ; conversely, if /zn(N) = 0for allw, then each of the functions gn <pN is v-negligible (13.14.1), hence so is /sup gn\q>N by (13.6.2), and so N is v-negligible. (13.15.9) Let (un) be an increasing sequence of positive measures having a least upper bound v in MR(X). For a function/ to be v-integrable it is necessary and Sufficient that f should be nn~integrable for all «, and that sup I \f\dpn< +00. n In that case we have Ifdv = lim /d/jn. J n-*o> J We may write fin — gn • v, where gn is ^0 and locally v-integrable, and the sequence (gn) is increasing. Moreover, we have v =/sup gn\ • v by (13.15.3), \ n J hence sup gn = 1 almost everywhere with respect to v. If/is v-integrable, then n so is/0,, for all /j, by virtue of (13.9.6), the inequality \fgn\ <; |/| (which is true almost everywhere with respect to v), and (13.9.13). Hence by (13.14.3) /is /V-integrable for all n. Conversely, if/is ^n-integrable for all n, then the func- tions^, are v-integrable (13.14.3) and we have/= lim^n almost every- -oo where with respect to v, so that / is v-measurable (13.9.11). Also we have I/I = sup \fgn\ almost everywhere with respect to v. Hence the result follows n from (13.9.13) and (13.8.1), and the last assertion of the proposition follows from the dominated convergence theorem (13.8.4). v). Furthermore, if v' is another positive measure on X